Problem: Integrate. $ \int 2\csc(x)\cot(x)\,dx $ Choose 1 answer: Choose 1 answer: (Choice A) A $-2\csc x + C$ (Choice B) B $-2\sec x + C$ (Choice C) C $-\sec x + C$ (Choice D) D $-\csc x + C$
Solution: We need a function whose derivative is $2\csc(x)\cot(x)$. We know that the derivative of $\csc(x)$ is $-\csc(x)\cot(x)$, so let's start there: $\dfrac{d}{dx} \csc(x) = -\csc(x)\cot(x)$ Now let's multiply by $-2$ : $\dfrac{d}{dx}\left[ -2\csc(x)\right] = -2\dfrac{d}{dx}\csc(x) =2\csc(x)\cot(x)$ Because finding the integral is the opposite of taking the derivative, this means that: $ \int 2\csc(x)\cot(x)\,dx =-2 \csc(x)\, + C$ The answer: $-2 \csc(x)\, + C$